Transcript ph201_overhead_ch6-sum07

Phy 201: General Physics I
Chapter 6: Work & Energy
Lecture Notes
What is Energy?
• Energy is a scalar quantity associated with the state of an
object (or system of objects).
– Energy is a calculated value that appears in nature and whose total
quantity in a system always remains constant and accounted for.
– Energy is said to be “Conserved”
• Loosely speaking, energy represents the “fuel” necessary for
changes to occur in the universe and is often referred to as
the capacity to perform work.
• We can think of energy as the “currency” associated with the
“transactions” (forces!) that occur in nature.
– In mechanical systems, energy is “spent” as force transactions are
conducted.
– Alternatively, the exertion of force requires an “expenditure” of energy.
• The SI units for energy are called Joules (J)
– In honor of James Prescott Joule
Work
• Work describes the energy transferred to/from an object by
the exertion of a force
• Work is essentially a measure of useful physical output
F
s
F
s
Definition:
standard form: W = F s cos= Fs  cos
or
component form: W = Fx x + Fy y + Fz z
Notes:
1. SI Units for Work: N.m
Work = Effort x Outcome
2. Unit comparison: 1 J = 1 N m

Examples of Work
(constant vs. variable force)
Two forces (a constant force and a non-constant force)
move an object 1.5 m:
F
5N
Constant
force
F
5N
AREA
AREA
s = 1.5 m
Non-Constant
force
s
The area under the left graph is:
Fconstant.s = 5N x 1.5m = 7.5 N.m = W (work)
The area under the right graph is:
Faverage.s = 5N x 1.5m = 7.5 N.m = W (work)
s = 1.5 m
s
Kinetic Energy
The energy associated with an object’s state of motion
– Kinetic energy is a scalar quantity that is never negative in value
Definition: KE = 12 mv2
2
KE =
1
2
mv2x +
1
2
mv2y +
1
2
mv2z
KE = KEx + KEy + KEz

 m  
SI
units:
kg

  
 s  

Key Notes:
1. The kinetic energy for the x, y, and z components are additive
2. Kinetic energy is relative to the motion of observer’s reference
frame, since speed and velocity are as well.
3. An object’s kinetic energy depends more on its speed than its
mass
4. Any change in an object’s speed will affect a change in its kinetic
energy
2
m
5. Unit comparison: 1 J = 1 kg  
s
Work & Kinetic Energy
• The net work performed on an object is related to the net
force:
W = F s cos = mas cos
Net

Net



• When FNet  0, a change in state of motion & kinetic energy is
implied:
WNet = KEf - KEi = KE
Derivation: WNet = FNetx x + FNety y= m ax x + ay y 


from Ch 3: ax x = 12 v2xf -v2xi same for y
combining the above relations:
WNet = m  ax x + ay y 
WNet =
1
2



m v2xf -v2xi + 12 m v2yf -v2yi

WNet = KEx + KEy
WNet = KE
The Work-Energy Theorem!
Work Performed by Gravitational Force
For a falling body (no air drag):
Fg=mg
Wy = Fgy = -mgy since cos = 1
Wx = Fgx = 0 since cos = 0
Wg = Wx + Wy = -mgy
y
 Gravitational force only performs
work in the vertical direction:
1. Wg is + when y is –
2. Wg is - when y is +
What about gravitational force on
an incline?
Fg=mg
y
Work Performed during Lifting & Lowering
• Consider Joey “blasting” his pecs with a bench
press workout (assume vlift = constant).
mg
Given: mbar = 100 kg
mbar g = -980 N
y= 1m
y
Applying Newton’s 2nd Law:
FNet  FLift - mbar g = mbar a = 0
FLift = mbar g = 980 N
The Work performed:
Wg = mbar gy = -980 N  m
WLift = FLift y = 980 N  m
WNet = FNet y = WLift + Wg = 0
FLift
Energy & the Body
• Our bodies utilize energy to:
– Sustain cellular processes & maintain body temperature
– Overcome joint friction
– Produce motion
• In nutrition, energy is typically described in terms of Calories
(i.e. the “calories” labeled on a cereal box):
– These “nutritional” calories are actually kilocalories:
1 Calorie = 1 kcal = 1000 cal
– The scientific calorie is related to the SI unit of energy:
1 cal = 4.186 J
Question(s): The “average” person requires ~2000 kcal per day.
1. How many joules are in 2000 kcal?
2. How high could you climb with this amount of energy?
3. It turns out the body uses only ~10% of its consumed energy for
physical activity. How high could you actually climb after consuming
2000 kcal?
Power
• Power is a measure of work
effectiveness
• Power is the time rate of energy
transfer (work) due to an exerted
force:
P=
dW F  dr

dt
dt
W
Average Power: Pavg=
t
WNet
KE
Average Net Power: PNet =
=
t
t
SI units: The Watt (1 W = 1 J/s)
Note: Power is also related to Force & Velocity:
Favgs
W
Pavg =
=
= Favgv
t
t
Power (cont.)
The same work output can be performed at various power
rates.
Example 1: Consider 100 J of work output accomplished over 2
different time intervals:
100 J
100 J
100 J over 1 s: P=
100 J over 100 s: P=
 100 W
1 W
1s
100 s
Example 2a: An 900 kg automobile accelerates from 0 to 30
m/s in 5.8 s. What is the average net power?
m 2
m 2

900
kg
30

0

  s   s  
K-Ko
Pnet =

 6.98 104 W or 93.6 hp
t
5.8 s
1
2
Example 2b: At the 30 m/s, how much force does the road
exert on this vehicle? Use the same power as 2a.
P 6.98 104 W
P=F  v  6.98 10 W  F  
 2330 N
m
v
30 s
4
Conservative vs. Nonconservative Forces
Conservative Forces:
1.
When the configuration of a system is altered, a force performs work
(W1). Reversing the configuration of the system results in the force
performing work (W2). The force is conservative if: W1= -W2
2. A force that performs work independent of the path taken.
3. A force in which the net work it performs around a closed path is
always zero or:
Wnet = 0 J {for closed path}
Examples:
Gravitational force (Fg), Elastic force (Fspring), and Electric force (FE)
Nonconservative Forces:
1.
2.
The work performed by the force depends on the path taken
When the configuration of a system is altered then reversed, the net
work performed by the force is not zero: Wnet ≠ 0 J {for closed path}
3. Work performed results in energy transformed to thermal energy
Examples:
Air Drag (FDrag) and Kinetic friction (fkinetic)
Sliding on an Incline
Example: No friction (conservative force)
A 1 kg object (vo=5 m/s) travels up a 30o incline and back down.
1. The Wnet performed by Fg (up):
2
2
Wup= 12 1 kg 0 ms  - 5 ms    -12.5 J


2. The Wnet performed by Fg (down):
2
2
Wdown= 12 1 kg -5 ms  - 0 ms    12.5 J


3. The total Wnet performed: Wnet by F  -12.5 J + 12.5 J = 0 J
g
Example: With Kinetic Friction (nonconservative force)
A 1 kg object (vo=5 m/s) travels up a 30o incline and back down against a 1.7
N kinetic friction force. Note: Block will not travel up as far as previous example.
The Wup performed by fk (up):
Wf up = fk x = -1.70 N1.89 m = -3.21 J
The Wdown performed by fk (down):
Wf down = fk x = 1.70 N-1.89 m = -3.21 J
The total Wnet performed:
Wnet by f = Wf up + Wf down = - 3.21 J - 3.21 J = -6.42 J
Defining or Identifying a System
• A system is a defined object (or group of objects) that are considered distinct from
the rest of its environment
• For a defined system:
1. all forces associated strictly with objects within the defined system are deemed internal
forces
• Internal forces do not transfer energy into/out of the system when performing work
within the system
Example: The attractive forces that hold the atoms of a ball together. These forces are
ignored when applying Newton’s 2nd Law to the ball.
2. all forces exerted from outside the defined system are deemed external forces
• External forces transfer energy into/out of the system when performing work on a
system
Example: The gravitational force that performs work on a falling object (the system)
increases the ball system’s (kinetic) energy.
Note: When the ball and the earth are together defined as the system, the work
performed by the gravitational force on the ball does NOT transfer energy into the
system.
• The appropriate of a system determines when a force is considered internal or
external & can go a long way toward simplifying the analysis of a physics problem
• The total energy associated with a defined system:
Esystem = PE + KE + Ethermal + EInternal
Work Done on a System by External Forces
• For a defined system, external forces are forces that are not
defined within the system yet perform work upon the system
• External forces transfer energy into or out of a system:
con
NC
WExt = WExt
+ WExt
= Esystem
WExt = PE + KE + EInternal+ Ethermal
– Conservative external forces alter the U and K (a.k.a. the mechanical
energy) of a system:
con
WExt
= Esystem= PE+ KE
– Nonconservative external forces may alter the mechanical energy (U,
K) as well as the non-mechanical energy (Einternal and/or Ethermal ) of a
system:
NC
WExt
= Esystem= PE+ KE+ EInternal+ Ethermal
Conservation of Energy
• In general, the total energy associated with a system of
objects represents the complete state of the system:
ETot= PE + KE + EInternal+ Ethermal
• Work represents the transfer of energy into/out of a system:
W = Esystem= PE + KE + EInternal+ Ethermal
• For an isolated system, the total energy within a system
remains a constant value:
Esystem= PE + KE + EInternal+ Ethermal= constant
or, for any 2 moments:
Esystem= PE1 + KE1 + EInternal 1+ Ethermal 1= PE2 + KE2 + EInternal 2+ Ethermal 2
• Considering only the mechanical energy of the system:
Emech= PE1 + KE1= PE2 + KE2  KE = -PE
Conservation of
Mechanical Energy!
Deeper Thoughts on Cons. of Energy
• Physicists have identified by experiment 3 fundamental
conservation laws associated with isolated systems:
1. Conservation of Energy
2. Conservation of Mass
3. Conservation of Electric Charge
• Treated as accepted “facts”, these laws have allowed for
experimental predictions that would not have been foreseen
otherwise:
1. Conservation of Energy led to the discovery of the neutrino during
neutron decay within the atomic nucleus
2. Conservation of Mass is fundamental in the prediction of new
substance formed during chemical processes
3. Conservation of Electric Charge predicts the formation of neutrons
do to the collision of protons with electrons, a process called Electron
Capture.
• Considered as accepted “facts”, these laws have allowed for
experimental predictions that would not have been foreseen
otherwise.
Feynman on Energy
"There is a fact, or if you wish, a law, governing
natural phenomena that are known to date.
There is no known exception to this law - it is
exact so far we know. The law is called
conservation of energy [it states that there is a
certain quantity, which we call energy that does
not change in manifold changes which nature Richard Feynman
(1918-1988)
undergoes]. That is a most abstract idea,
because it is a mathematical principle; it says that there
is a numerical quantity, which does not change when
something happens. It is not a description of a
mechanism, or anything concrete; it is just a strange fact
that we can calculate some number, and when we finish
watching nature go through her tricks and calculate the
number again, it is the same...”
James Prescott Joule (1818-1889)
• English inventor & scientist
• Interested in the efficiency of electric motors
• Described the heat dissipated across a
resistor in electrical circuits (now known as
Joules’ Law)
• Demonstrated that heat is produced by the
motion of atoms and/or molecules
• Credited with establishing the mechanical
energy equivalent of heat
• Participated in establishing the “Law of
Energy Conservation”
"It is evident that an
acquaintance with
natural laws means no
less than
an acquaintance with
the mind of God therein
expressed."